Problem 11 - Differential Equations and Linear Algebra

Question

Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix A is not invertible, therefore there is no inverse for matrix A.

Step 1: Set up the augmented matrix [A|I]

: First, set up the augmented matrix with matrix A and the identity matrix I. $$ [A|I]=\left[\begin{array}{ccc|ccc} 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 & 0 & 1 \end{array}\right] $$

Step 2: Perform row operations

: Now, perform row operations to try to get the identity matrix on the left side of the augmented matrix. Since the first column has only zeros, we can move to the second column. 1. Swap row 1 and row 3: $$ \left[\begin{array}{ccc|ccc} 0 & 1 & 2 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \end{array}\right] $$ 2. Perform R3 = R3 - R1: $$ \left[\begin{array}{ccc|ccc} 0 & 1 & 2 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 & 0 & -1 \end{array}\right] $$ 3. Perform R3 = -1/2 R3: $$ \left[\begin{array}{ccc|ccc} 0 & 1 & 2 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1/2 & 0 & 1/2 \end{array}\right] $$ 4. Perform R1 = R1 - 2R3 and R2 = R2 - R3: $$ \left[\begin{array}{ccc|ccc} 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1/2 & 1 & -1/2 \\ 0 & 0 & 1 & -1/2 & 0 & 1/2 \end{array}\right] $$ Since we reached a row with all zeros except for the identity matrix, it means the matrix A is not invertible. Therefore, there is no inverse for matrix A.

Key Concepts

Inverse Matrix

The inverse of a matrix, denoted as \(A^{-1}\), is essentially the "reverse" of a matrix. When you multiply a matrix by its inverse, you should get the identity matrix, symbolized by \(I_n\). This is similar to multiplying a number by its reciprocal to get one. Not every matrix has an inverse. For a matrix to have an inverse, it must be square (the number of rows must equal the number of columns) and its determinant must be non-zero. If a matrix does not meet these criteria, it is called singular, indicating it cannot be inverted. In the Gauss-Jordan method, if you cannot transform a matrix into an identity matrix through row operations, it means the matrix doesn't have an inverse.

Augmented Matrix

An augmented matrix combines two matrices into one, often during processes like finding inverses or solving systems of linear equations. Essentially, you append matrices side by side. For example, to find an inverse using the Gauss-Jordan method, you create an augmented matrix \([A|I]\) by placing the identity matrix \(I\) beside the matrix \(A\). This setup allows you to perform various row operations simultaneously on \(A\) and \(I\). The idea is to use techniques like row swapping, scaling, and addition/subtraction to convert \(A\) into an identity matrix on the left. If successful, the other side becomes the inverse matrix of \(A\). However, if it is not possible to achieve this, as in the case above, \(A\) has no inverse.

Identity Matrix

The identity matrix acts as the "1" in the world of matrices. It is a square matrix with ones on the diagonal from the top left to the bottom right and zeros elsewhere. For any given size \(n\), the identity matrix \(I_n\) looks like this in a 3x3 form:

• \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\)

Using the identity matrix, you can test whether an inverse is correctly calculated since multiplying a matrix by its inverse should result in the identity matrix. Its presence is crucial in many mathematical operations, serving as a neutral element in matrix multiplication. During the Gauss-Jordan process, reaching a state where the original matrix is converted to an identity matrix indicates that the process to find the inverse was effective, except, of course, when transformations produce contradictions leading to a conclusion that an inverse does not exist.